Riemann Function Research Articles

Boundary Value Problems for a Model Hyperbolic Equation of the Third Order with Non-local Conditions of the First Kind

Considers nonlocal problems of the first kind for a model third-order hyperbolic equation. The main goal of the article is to prove the solvability of nonlocal problems of the first kind for a model third-order hyperbolic equation. Using the Riemann function method, the problem is reduced to Volterra integral equations of the second kind. Using the method of integral equations, the existence of a unique solution to nonlocal problems of the first kind is proven. The resulting solution to nonlocal problems of the first kind makes it possible to describe the process of moisture transfer in soils, heat transfer in a heterogeneous medium, and fluid filtration in porous media.

Discover an accurate approximation of dynamical system without prior information and customized design

Modeling dynamical systems is fraught with challenges when data can be collected but thorough analysis of the mechanism is absent. We design a method to discover unknown dynamical systems from data. The method discovers an accurate approximation of the model without the prior information and the customized design for each problem. The identification steps are straightforward as bringing in the data and then obtaining the model. The method begins with the simple idea that the equations of motion of many practical problems are Riemann integrable functions. For this reason, the Fourier series can decompose the equations of motion. In order to improve the accuracy, we design an extension that helps us to approximate unknown functions by the Fourier series with a high rate of convergence. The idea converts the difficulty of modeling the dynamical system into finding its Fourier series approximation. Convenient procedures enable the modeling of different problems. Numerical examples show that the new method discovers linear and nonlinear dynamical systems in the same steps and without the prior information.

Formation of singularities in plasma ion dynamics

We study the formation of singularity for the Euler–Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic equations fail to exist globally in time. We establish criteria for C 1 blow-up of the Euler–Poisson system, both for the isothermal and pressureless cases. In particular, our blow-up condition for the pressureless model does not require that the gradient of velocity is negatively large. In fact, our result particularly implies that the smooth solutions can break down even if the gradient of initial velocity is trivial. For the isothermal case, we prove that smooth solutions leave C 1 class in a finite time when the gradients of the Riemann functions are initially large.

An Analogue of the Darboux Problem for Hyperbolic Equations of the Third Order in a Rectangular Triangular Domain

The article considers the Darboux problem for a third-order hyperbolic equation in a rectangular triangular domain. The main purpose of the article is to prove the solvability of the Darboux problem. Using the Riemann function method, the problem is reduced to Volterra integral equations of the second kind. The existence of a unique solution to the Darboux problem is proved by the method of integral equations. The obtained solution of the Darboux problem allows us to describe the process of moisture transfer in soils, heat transfer in a heterogeneous medium, and liquid filtration in porous media.

The pointwise behavior of Riemann’s function

We present a new and simple method for the determination of the pointwise Hölder exponent of Riemann's function \sum_{n=1}^{\infty} n^{-2}\sin(\pi n^{2} x) at every point of the real line. In contrast to earlier approaches, where wavelet analysis and the theta modular group were needed for the analysis of irrational points, our method is direct and elementary, being only based on the following tools from number theory and complex analysis: the evaluation of quadratic Gauss sums, the Poisson summation formula, and Cauchy's theorem.

Generalized Riemann Functions, Their Weights, and the Complete Graph

By a Riemann function we mean a function $f:\mathbb{Z}^n\to\mathbb{Z}$ such that $f(\mathbf{d})$ is equals $0$ for $d_1+\cdots+d_n$ sufficiently small, and equals $d_1+\cdots+d_n+C$ for a constant, $C$, for $d_1+\cdots+d_n$ sufficiently large. By adding $1$ to the Baker-Norine rank function of a graph, one gets an equivalent Riemann function, and similarly for related rank functions.
 To each Riemann function we associate a related function $W: \mathbb{Z}^n\to \mathbb{Z}$ via Möbius inversion that we call the weight of the Riemann function. We give evidence that the weight seems to organize the structure of a Riemann function in a simpler way: first, a Riemann function $f$ satisfies a Riemann-Roch formula iff its weight satisfies a simpler symmetry condition. Second, we will calculate the weight of the Baker-Norine rank for certain graphs and show that the weight function is quite simple to describe; we do this for graphs on two vertices and for complete graphs.
 For complete graphs, we build on the work of Cori and Le Borgne who gave a linear time method to compute the Baker-Norine rank of the complete graph. The associated weight function has a simple formula and is extremely sparse (i.e., mostly zero). Our computation of the weight function leads to a new linear time algorithm to compute the Baker-Norine rank, via a new formula likely related to one of Cori and Le Borgne, but seemingly simpler for general $\mathbf{d}\in \mathbb{Z}^n$, namely$$r_{{\rm BN},K_n}(\mathbf{d}) =-1+\biggl| \biggl\{ i=0,\ldots,\deg(\mathbf{d}) \ \Bigm|\ \sum_{j=1}^{n-2} \bigl( (d_j-d_{n-1}+i) \bmod n \bigr) \le \deg(\mathbf{d})-i\biggr\} \biggr|.$$However, the formula of Cori and Le Borgne, which requires $\mathbf{d}\in \mathbb{Z}^n$ to be a sorted parking function, is easier to evaluate for such $\mathbf{d}$.
 Our study of weight functions leads to a natural generalization of Riemann functions, with many of the same properties exhibited by Riemann functions.

The Quantized Constants with Remmen’s Scattering Amplitude to Explain Riemann Zeta Zeros

Riemann Hypothesis has been proposed by Bernhard Riemann since year 1859. Nowadays, there are lots of proof or disproof all over the internet society or the academic professional authority etc. However, none of them is accepted by the Clay’s Mathematics Institute for her Millennium Prize. In the past few months, this author discovered that there may be a correlation exists between the real and imaginary parts of Riemann Zeta function for the first 10 non-trivial zeros of the Riemann function etc. Indeed, when one tries to view the correlation relationship as a constant like the Planck’s one. Then we may show that Riemann Zeta zeros are indeed discrete quantum energy levels or the discrete spectrum as electrons falling from some bound quantum state to a lower energy state (or Quantum Field Theory). That may be further explained by Remmen’s scattering amplitude or the S-matrix. We may approximate the S-matrix by applying the HKLam theory to it and predict the scattering amplitude or even the Riemann Zeta non-trival zeros etc. By the way, the key researching equations or formula in the following content will be around the Taylor expansion of the Riemann Zeta function, their convergence etc. In additional, I will also investigate the (*’’) as shown below: ∏_(i=1)^∞▒(z-z_i ) = ξ(0.5 + i*t) = (∑_(n=1)^∞▒1)⁄n^((0.5+i*t) ) = ∏_(j=1)^∞▒(1-1⁄(p_j^((0.5+i*t) ) ))^(-1) ------------ (*’’) as we may find the existence of some constants like the Planck’s one. For the application of the aforementioned scholarly outcome, it is well-known that if one can find the pattern of the appearance to the prime number and hence break the public key cryptography in the everyday usage of information technology security etc.

A Statistical Method for the Prediction of the Bear and Bull Stock Markets Based on the Zeroes of Riemann’s Zeta Function

We define a method for predicting the stochastic behavior of the Bull and Bear periods of the stock market. In this paper, initially, we carry on a comprehensive evaluation of more frequently used statistical methods for evaluating Stock markets. Our work is based on collecting 40 years of data from the Italian stock market. The proposed solution is defined using the statistical analysis of the Bear and Bull Stock markets. We defined a new system to predict the trend of a stock market price, where the trend of the succession of Bull and Bear markets can be described by a probability density function given by a Gaussian distribution. Furthermore, we consider the inverses of the relative time intervals as a measure of the speed with which the phenomenon of the Bear market (or, equivalently, the Bull market) develops in that interval of time. Therefore, this factor can ultimately represent the first statistical weight of the single percentage variation. Again, the time intervals of the individual Bear and Bull market periods are considered, calculated from 01/01/1973. This allows us to consider the hypothesis that a secondary factor of probability is the temporal distance of the event that has already occurred. This work includes a criterion for statistically generating the most probable values of the next Bear and Bull markets and the length of the time intervals corresponding to these market situations. This criterion is based on the following hypothesis: 
 To obtain the distribution of the predictive points of max and min in the succession of Bull and Bear markets, it is assumed that, in the long period, the random distribution of the successive max and min takes the trend of the distribution of the distance fluctuations between the zeroes of the Riemann's function which, in turn, is approximated by a Unitary Gaussian Distribution (GUE). Our results show that: 
 
 The linear interpolation of the Variations of the market (positive and negative) relative to different and successive sampling sets for future trends do not show high percentage variations between them, 
 Above all, the lengths of the single time intervals of the future variations, relative to different and successive sampling sets, are quite similar to each other. Hence, the method appears to be basically stable and promising. 

A modular relation involving a generalized digamma function and asymptotics of some integrals containing Ξ(t)

A modular relation of the form $F(\alpha, w)=F(\beta, iw)$, where $i=\sqrt{-1}$ and $\alpha\beta=1$, is obtained. It involves the generalized digamma function $\psi_w(a)$ which was recently studied by the authors in their work on developing the theory of the generalized Hurwitz zeta function $\zeta_w(s, a)$. The limiting case $w\to0$ of this modular relation is a famous result of Ramanujan on page $220$ of the Lost Notebook. We also obtain asymptotic estimate of a general integral involving the Riemann function $\Xi(t)$ as $\alpha\to\infty$. Not only does it give the asymptotic estimate of the integral occurring in our modular relation as a corollary but also some known results.

The boundary between the Prime Gaps and the Interchangeable be-tween a Quantum Mechanics System & a Classical Chaos One

With reference to my last paper, this author found that there are always some discrete quantum constants like the Planck’s one laying between the real and the imaginary parts of the Riemann function (I.e when we fix or group the Riemann Zeros one by one and make the real part to be varying.) or the quantum mechan-ics system. Hence, the Riemann Zeta Zeros give us the discrete quantum con-stants together with the given discrete quantum constants may give you the wanted Riemann Zeta Zeros. This phenomenon may be explained by Quantum Riemann Zeta Epiphancy. In the present paper, this author tries to fix or group the real part of the Riemann Zeros and make the imaginary part of the Riemann Zeros to be varying from one to the tenth one or resulting a chaos like graph. Thus, it seems that we may interchange a quantum mechanics system with the classical chaos one by just varying one variable and fix another one (such as varying real part of the Riemann function’s zeors from 0.1 to 1 and fix the imaginary part of the Rie-mann zeros one group after one another). Certainly, the vice versa for the fix of the real part (one group after another) with different Riemann Zeros may also help. In other words, the quantum mechanics system is just the conjugate pair (varying x and fixed y variables) of the classical chaos one or the vice versa is al-so true (fixed x and varying y variables) & put back into the Zeta functions etc. Hence, it is a method (or transformation) for us so that we may easily to make a interchange between these two systems by only adjusting or selecting which vari-ables to be varying or fixed. Then we may turn the classical chaos system into a quantum mechanics one or the vice versa. This implies that it may be possible to apply controllable quantum systems for the investigation of our natural aspects. Finally, this author have also shown that there is a boundary laying be-tween the prime gap through the elementary calculation and mathematical deriva-tion without using the computer software etc. It is thus a contradiction to the fact that there is no bound for the prime gap. This tells us that there is a need to have a shift from zero to 0.5 for the real part of the Riemann Zeta Zeros as what this author has mentioned in my last paper etc.

The Riemann function of the Cauchy problem for a second-order hyperbolic equation with a growing coefficient

The Riemann function of the Cauchy problem for a second-order hyperbolic equation with a growing coefficient

Boundary Value Problems for a Mixed Fourth-order Parabolic-Hyperbolic Equation With Discontinuous Gluing Conditions

The theorem of the existence and uniqueness of the solution of the boundary value problem for the equation in partial derivatives of the fourth order with variable coefficients containing the product of the mixed parabolic-hyperbolic operator and the differential operator of the oscillation string with discontinuous conditions of gluing in the pentagon to the plane is proved. By the method of reducing the order of equations, the solvability of the boundary value problem is reduced to the solution of the Tricomi problem for the mixed parabola-hyperbolic equation with variable coefficients and discontinuous gluing conditions. Solving this problem is reduced to the solution of Fredholm’s integral equation of the second order relative to the trace of the derivative function on y along the line of variation of the equation type. In the hyperbolic part of the domain, the representation of the solution of the problem for the hyperbolic equation with the smallest terms was obtained by using the Riemann function method. In the parabolic part of the domain, the solution of the first boundary value problem for the parabolic equation with the smallest terms is obtained by the method of successive approximations and the Green’s function. As a result, the solution of the problem is realized by the method of solving the Gursa problem and the first boundary value problem for the equation of string oscillation.

GOURSAT PROBLEM FOR THE ADJOINT MODEL TELEGRAPH EQUATION WITH THE RATE a(s, t) IN THE UPPER HALF-PLANE

In the upper half-plane, the classical solution and correctness criterion of the Goursat problem for a linear inhomogeneous adjoint model telegraph equation with variable rate a(s,t) are found explicitly. An explicit formula is obtained for the classical solution of this Goursat problem, unique and stable with respect to the right hand side of the equation and Goursat data. This formula contains implicit characteristic functions of the equation. In the case of a homogeneous conjugate model telegraph equation, the classical solution of this Goursat problem is the Riemann function in all linear mixed (initial-boundary) problems for an inhomogeneous model telegraph equation with variable rate a(s,t). This Riemann function has been calculated by us. A correctness criterion according to Hadamard (necessary and sufficient conditions) of its unique and stable on the right-hand side of the equation and the Goursat data solvability is found. This criterion consists of smoothness requirements on the righthand side of the equation and two Goursat data. The smoothness requirements on the right side of the equation are the condition of continuity of the right-side and the corresponding integral smoothness conditions on the right side of the equation and on the Goursat data – their twice continuous differentiability in the upper half-plane.

On an Integral Equation with the Riemann Function Kernel

This paper is concerned with a study of a special integral equation. This integral equation arises in many applied problems, including transmutation theory, inverse scattering problems, the solution of singular Sturm–Liouville and Shrödinger equations, and the representation of solutions of singular Sturm–Liouville and Shrödinger equations. A special integral equation is derived and formulated using the Riemann function of a singular hyperbolic equation. In the paper, the existence of a unique solution to this equation is proven by the method of successive approximations. The results can be applied, for example, to representations of solutions to Sturm–Liouville equations with singular potentials, such as Bargmann and Miura potentials, and similiar. The treatment of problems with such potentials are very important in mathematical physics, and inverse, scattering and related problems. The estimates received do not contain any undefined constants, and for transmutation kernels all estimates are explicitly written.

Strong derivative and the essentially Riemann integral

We characterize the primitives of Riemann integrable functions using the concept of strong differentiability. We also introduce the notion of essentially Riemann integrable function and characterize it descriptively using its primitive. We also give a characterization of this function that parallels Lebesgue's criterion for Riemann integrability.

On the Hausdorff dimension of Riemann’s non-differentiable function

Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann's function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions.

A series representation for Riemann's zeta function and some interesting identities that follow

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta(s)$, and hence Riemann's function $\zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(i\kappa)$ of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions $\zeta^{\pm}(s)$ and $\eta^{\pm}(s)$ are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints.

A Speedy New Proof of the Riemann's Hypothesis

In this paper we show that Riemann's function <i>(xi)</i>, involving the Riemann’s (<i>zeta)</i> function, is holomorphic and is expressed as a convergent infinite polynomial product in relation to their zeros and their conjugates. Our work will be done on the critical band in which non-trivial zeros exist. Our approach is to use the properties of power series and infinite product decomposition of holomorphic functions. We take inspiration from the Weierstrass method to construct an infinite product model which is convergent and whose zeros are the zeros of the zeta function. By applying the symetric functional equation of the <i>xi</i> function we deduce a relation between each zero of the function <i>xi</i> and its conjugate. Because of the convergence of the infinite product, and that the elementary polynomials of the second degree of this same product are irreducible into the complex set, then this relation is well determined. The apparent simplicity of the reasoning is based on the fundamental theorems of Hadamard and Mittag-Leffler. We obtain the sought result: the real part of all zeros is equal to ½. This article proves that the Riemann’ hypothesis is true. Our perspectives for a next article are to apply this method to Dirichlet series, as a generalization of the Riemann function.

Fourier Transform Associated with Riemann Zeta Function

The Fourier transform associated with the normalized logarithm of the modulus of the Riemann Zeta Function is considered. The formulas linking the Fourier transform and the zeros of the Riemann function are established that lead to the necessary and sufficient condition of satisfaction of the Riemann hypothesis.

О построении функции Римана-Адамара для трехмерного уравнения Бианки

In this paper, we state the Darboux problem and give the definition of the Riemann–Hadamard function for a third-order equation with dominant partial derivative (the Bianchi equation). Basing on the possibility of explicitly representing the Riemann function for a certain class of Bianchi equations of the third order, which are equivalent with respect to function, we establish sufficient conditions for coefficients of the Bianchi equation that provide the construction of the Riemann–Hadamard function in terms of hypergeometric functions.